question 1 this question concerns the volume of an object bounded by a
This question concerns the volume of an object bounded by a parabola x = y² - 5y on the left and a
straight line x = y on the right (z and y in metres). The cross section is show below:
x = y? — 5y
The height of the object, h(x, y) (in metres), is modelled in this assessment using the following function:
h(x,y) = (y − z)(z − y? +5y).
(a) (5 marks) Find the maximum height by doing the following:
i) find all the critical points of the function h(x, y),
ii) identify the single critical point that is within the cross section area A depicted above (not
including points on the boundary).
iii) show that this critical point is a local maximum using the Hessian determinant test,
iv) evaluate the value of h at this maximum
(b) (5 marks) The volume of the object V (metres³) is defined as the integral
=ff h(x, y) da
where A is the cross-section depicted in the figure above. Calculate the volume by doing the
following:/n(c) (2 marks) use MATLAB to create a contour plot of the height h(x, y). Make sure to include a few
(three or more) contours (level curves) between zero and the maximum value you found in Qla.
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